Magnetic reconnection in high-lundquist-number plasmas. N. F. Loureiro Instituto de Plasmas e Fusão Nuclear, IST, Lisbon, Portugal

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1 Magnetic reconnection in high-lundquist-number plasmas N. F. Loureiro Instituto de Plasmas e Fusão Nuclear, IST, Lisbon, Portugal Collaborators: R. Samtaney, A. A. Schekochihin, D. A. Uzdensky 53 rd APS DPP Meeting, 2011

2 Resistive MHD reconnection: an unsolved problem Resistive MHD is the simplest framework with which reconnection can be described. Until ~5 years ago, it was believed that resistive MHD reconnection was accurately described by the Sweet-Parker (SP) model: S = L CS V A /η δ SP /L CS S 1/2 u in /V A S 1/2 ce B 0 V A S 1/2 L CS δ SP In most systems of interest, S>>1: SP model is too slow to explain observations. However, collisionless reconnection is fast.

3 Resistive MHD reconnection: an unsolved problem However: 1. Not all plasmas are collisionless! Many astrophysical environments are very high density solar chromosphere, interstellar medium, inside stars and accretion disks, etc so reconnection layer is collisional and resistive MHD should apply. Can reconnection be fast in these environments?

4 Resistive MHD reconnection: an unsolved problem However: 1. Not all plasmas are collisionless! Many astrophysical environments are very high density solar chromosphere, interstellar medium, inside stars and accretion disks, etc so reconnection layer is collisional and resistive MHD should apply. Can reconnection be fast in these environments? 2. Numerical evidence for current sheet instability to secondary islands (plasmoids) has been around for quite a while (e.g., Park 84, Steinolfson 84, Biskamp 86, Loureiro 05) (Loureiro et al., PRL 05)

5 Resistive MHD reconnection: an unsolved problem However: 1. Not all plasmas are collisionless! Many astrophysical environments are very high density solar chromosphere, interstellar medium, inside stars and accretion disks, etc so reconnection layer is collisional and resistive MHD should apply. Can reconnection be fast in these environments? 2. Numerical evidence for current sheet instability to secondary islands (plasmoids) had been around for quite a while (e.g., Park 84, Steinolfson 84, Biskamp 86, Loureiro 05) 3. Background turbulence may drastically affect the reconnection process, make it fast (Matthaeus & Lamkin 86, Lazarian & Vishniac 99, Kowal et al., 09, Loureiro et al., 09)

6 SP current sheet instability (Loureiro et al., 07) Rigorous analytical linear theory. Main points: 1- Assume incompressible flow profile of the form: u x = v A x/l CS ; u y = V A y/l CS

7 SP current sheet instability (Loureiro et al., 07) Rigorous analytical linear theory. Main points: 1- Assume incompressible flow profile of the form: u x = v A x/l CS ; u y = V A y/l CS 2- Obtain consistent reconnecting magnetic field from resistive induction equation.

8 SP current sheet instability (Loureiro et al., 07) Rigorous analytical linear theory. Main points: 1- Assume incompressible flow profile of the form: u x = v A x/l CS ; u y = V A y/l CS 2- Obtain consistent reconnecting magnetic field from resistive induction equation. 3- Linearize RMHD eqs and look for perturbations γ V A /L CS 1/τ A

9 SP current sheet instability (Loureiro et al., 07) Rigorous analytical linear theory. Main points: 1- Assume incompressible flow profile of the form: u x = v A x/l CS ; u y = V A y/l CS 2- Obtain consistent reconnecting magnetic field from resistive induction equation. 3- Linearize RMHD eqs and look for perturbations γ V A /L CS 1/τ A 4- Asymptotic expansion in: δ CS /L CS 1 kl CS κ 1 κ 1

10 SP current sheet instability (Loureiro et al., 07) Rigorous analytical linear theory. Main points: 1- Assume incompressible flow profile of the form: u x = v A x/l CS ; u y = V A y/l CS 2- Obtain consistent reconnecting magnetic field from resistive induction equation. 3- Linearize RMHD eqs and look for perturbations 4- Asymptotic expansion in: 5- Obtain: γ V A /L CS 1/τ A γ max τ A S 1/4 k max L CS S 3/8 δ CS /L CS 1 kl CS κ 1 κ 1

11 SP current sheet instability (Loureiro et al., 07) Plasmoids galore!! Rigorous analytical linear theory. Main points: 1- Assume incompressible flow profile of the form: u x = v A x/l CS ; u y = V A y/l CS 2- Obtain consistent reconnecting magnetic field from resistive induction equation. 3- Linearize RMHD eqs and look for perturbations 4- Asymptotic expansion in: 5- Obtain: Super Alfvenic growth!! γ V A /L CS 1/τ A γ max τ A S 1/4 k max L CS S 3/8 δ CS /L CS 1 kl CS κ 1 κ 1

12 Current sheet instability: threshold Linear theory predicts: To a good approximation, outflows in the CS are linear (Yamada et al. 00, Uzdensky & Kulsrud 00): v y V A y /L CS γ max τ A ~ S 1/ 4 k max L CS ~ S 3 / 8 For any perturbation to grow, its growth rate needs to exceed the shearing rate: γτ A >>1 S 1/ 4 >>1 Critical threshold for instability: S c ~ 10 4

13 Numerical confirmation of linear theory Numerical simulations confirm scalings predicted by linear theory (Samtaney et al., PRL 09). (independently confirmed by Huang et al., PoP 10)

14 Hierarchical Plasmoid Chains Long current sheets (S > S c ~ 10 4 ) are violently unstable to multiple plasmoid formation. Current layers between any two plasmoids are themselves unstable to the same instability if S n = L n V A /η >S c Plasmoid hierarchy ends at the critical layer: (Shibata-Tanuma 01) N ~ L / L c plasmoids separated by nearcritical current sheets.

15 Plasmoid-dominated reconnection: the ULS model New theoretical model (ULS) (Uzdensky et al., PRL 10) Key assumptions: 1. X-point collapse and layer instability sufficiently fast so any 2 plasmoids separated by critical layer (cf. Waelbroeck 93, Loureiro 05) 2. Upstream magnetic field at any level B 0 ~V A. 3. Plasmoids do not saturate before ejection from their respective layers (can be proven rigorously a posteriori)

16 Plasmoid-dominated reconnection: the ULS model New theoretical model (ULS) (Uzdensky et al., PRL 10) Key results: Nonlinear statistical steady state exists; effective reconnection rate is E eff ~ S c -1/2 ~ 0.01 independent of S! Plasmoid flux and size distribution functions are: f(ψ) ~ ψ -2 ; f(w x ) ~ w x -2 (because ψ~w x B 0 ) Monster plasmoids form occasionally: w max ~ 0.1 L --- can disrupt the chain, observable

17 Numerical results Direct numerical simulations to investigate magnetic reconnection at S>S c and test assumptions and predictions of ULS model --- Loureiro et al., arxiv: Typical snapshot. S=10 6, domain size: Lx=0.3L; Ly=0.5L; res

18 Reconnection and dissipation rates Ẽ eff 0.02 ~ 40% of incoming magnetic energy dissipated into heat

19 Plasmoid flux distribution function

20 Plasmoid width distribution function

21 Width vs. Flux Distribution Function Diagonal (bold): ULS plasmoids. BUT, there s also a significant off-diagonal component

22 Width vs. Flux Distribution Function Consider the coalescence between two plasmoids. ULS assumed it to be instantaneous. In reality it takes a finite time.

23 Width vs. Flux Distribution Function i.e., off-diagonal

24 Width vs. Flux Distribution Function When does coalescence matter for the distribution function? t cl Ψ 0 /ce B 0 w x0 /ce < τ A

25 Monster plasmoid formation

26 Monster matters Probability of formation of a monster plamoid: P M = L w M f(w x )dw x In our simulations: P M ~ 1%--3%, independent of S! i) The entire plasmoid populations is renewed in ii) Avg. number of plasmoids in the sheet at any instant: t ej τ A N L/L c S/S c Time to form a monster τ M /τ A 1+(1/P M )(S c /S) Thus, for S > S c /P M ~ 10 6, τ Μ τ Α --- independent of S!!

27 Time to monster independent of S

28 Conclusions Current sheets predicted by the SP theory are violently unstable to the formation of high wavenumber plasmoid chains (Loureiro 07, Samtaney 09). Plasmoids disrupt the original current-sheet and give rise to multi-layered, fractal-like, turbulent structure. MHD reconnection in large S systems is fast, dynamic, bursty. Sweet-Parker theory inadequate. Plasmoid-dominated reconnection requires statistical description. The ULS model (Uzdensky 10) is the first attempt at describing this new type of MHD turbulence. Numerical simulations (Loureiro et al., arxiv: ) confirm and partially amend the theoretical predictions of the ULS model. Both the reconnection and the dissipation rates asymptote to S-independent values. ~ 40% of incoming energy dissipated into heat. Plasmoid flux/width distribution functions scale as f(ψ) ~ ψ -2 ; f(w x ) ~ w x -2 ; coalescence gives rise to off-diagonal plasmoids. Monster plasmoids form with probability ~1%--3%, independent of S; time to form them is ~ the ejection time, ~ τ A, also independent of S.